On the tidal heating of Enceladus P. Varga, B. Süle, E. Illés-Almár To cite this version: P. Varga, B. Süle, E. Illés-Almár. On the tidal heating of Enceladus. Journal of Geodynamics, Elsevier, 2009, 48 (3-5), pp.247. 10.1016/j.jog.2009.09.031. hal-00594415 HAL Id: hal-00594415 https://hal.archives-ouvertes.fr/hal-00594415 Submitted on 20 May 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Accepted Manuscript Title: On the tidal heating of Enceladus Authors: P. Varga, B. Sule,¨ E. Illes-Alm´ ar´ PII: S0264-3707(09)00098-2 DOI: doi:10.1016/j.jog.2009.09.031 Reference: GEOD 925 To appear in: Journal of Geodynamics Please cite this article as: Varga, P., Sule,¨ B., Illes-Alm´ ar,´ E., On the tidal heating of Enceladus, Journal of Geodynamics (2008), doi:10.1016/j.jog.2009.09.031 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. On the tidal heating of Enceladus P.Varga(1)*, B.Süle(1) ,E.Illés-Almár(2) (1) Geodetic and Geophysical Research Institute, Seismological Observatory, Budapest, Meredek u. 18 H-1112, Hungary ([email protected]) (2) Konkoly Observatory, Budapest, Hungary, H-1525, P.O.B. 67 Abstract Enceladus, one of Saturn’s moons, shows significant volcanic activity identified by the Cassini spacecraft. The aim of the present study is to investigate - with the adaptation of mathematical tools used in geodynamics - the extent of tidal heating due to the mean motion resonance with Dione. For the purpose of calculations a two-layer model of Enceladus was used. The inner part of the model is a “rocky core” with a relative radius 0.55, while the outer part is composed of water ice. The results of model calculations show that the effective tidal heating is not uniformly distributed within Enceladus. It was found for the selected model of Enceladus, that the tidal heating is maximum within the depth interval (25-75) km. Due to the inhomogeneity within Enceladus, 85% of the tidal energy is generated in a volume that contains just 39% of its mass In time intervals of 3.0x108 and 5.3x108 years the temperature increase in the relative depth range 0.70≤r/aE ≤0.90 is approximately 270° and 370° Kelvin, respectively. Keywords: Saturn, Enceladus, tide, solid body 1.Introduction The medium size, spherical satellites of Saturn (listed in Table 1) were probably locked into synchronized rotation states with Saturn early in their history. Consequently, their tidal bulge has no motion in a coordinate system fixed to the body of the moon. Under this condition there is no tidal friction influencing the axial spin. However, this situation changes if the orbital motion of a moon is also in resonance with the orbital motions of other members of the satellite system. This resonance generates a forced eccentricity that leads to temporal variation of the tidal bulge due to gravitation from the central body of the system (Peale & Cassen,1979; Wisdom, 2004). Enceladus (mean radius aE=252.1 km) orbits around Saturnbetween Mimas and Tethys. Its orbit has evolved into mean motion resonance with Dione which generates its orbital eccentricity 0.0045 (Table 1). Enceladus is probably the brightest object in the solar system, its water ice surface reflecting almost 100% of the sunlight (Spencer et al., 2006). Contrary to its neighbours Mimas and Thetys the surface of Enceladus shows the presence of both internal and external tectonic processes. Cryovolcanism almost certainly has resurfaced Enceladus at various epochs of its history (Peale, 2003). According to Ross and Schubert (1989) the geologic activity of resurfaced regions could be as old as 1.0-1.7 Gyr, which suggests a similar age for the presence of orbital resonance. An essential part of Enceladus is covered by almost crater-free plains with north-south trending fractures with Y-shaped discontinuities (Porco et al.2006). Similar linear structures were observed on the surface of the Earth in the the cosmo- geological map of the former Soviet Union (Brjukanov et al. 1984). This fault system on the surface of the Earth is not caused by recent tectonic activity, has a regular distribution with respect to theAccepted axis of rotation and has been expl ainedManuscript by stress fields of a despinning planet (Melosh 1977). No object in the solar system smaller than Enceladus is known to be tectonically active. Traces of recent eruptions on Enceladus were detected already by Squyres et al. (1983) in ---------------------------------------- *Corresponding author. Phone:+36-1-248 2321, Fax:+36-1-248 2301, e-mail:[email protected] 1 Page 1 of 11 form of the resurfacing of fresh materials on the surface. Plumes erupting from the south polar region of Enceladus have been detected in pictures taken by the Cassini spacecraft from 16 January 2005 (Porco et al. 2006). A fountain higher than 435 km was detected in Cassini images observed on 25 November 2005. The elevated South Pole temperature anomaly detected by Cassini, connected with plumes emanating from this region, corresponds closely to a recently active region of four linear depressions called “tiger stripes”. This circumstance renders probable that these structures are deep enough to conduct the water vapour from the interior of Enceladus to the surface. Very recently Hurford et al. (2007) “…report a mechanism in which temporal variations in tidal stress open and close the tiger stripe rifts, governing the timing of eruptions. During each orbit, every portion of each tiger stripe rift spends about half the time in tension, which allows the rift to open, exposing volatiles and allowing eruptions”. This phenomenon requires large subsurface tidal stresses that imply the presence of a liquid ocean. To explain this phenomenon of recent tectonic activity a heat source within Enceladus is needed. Possibilities are: - tidal heating associated with eccentricity of the orbit - tidal heating due to the librations of Enceladus - presence of radioactive heating. The main goal in this paper is to show to what extent the tidal heating can melt the outer icy part of Enceladus. The frictional energy due to the tides can be dissipated as heat. The released heat from the south polar tiger stripes region has been estimated to be (5.8±1.9) GW (Spencer et al., 2006). Contributions published until now were not able to establish a complete theoretical model able to produce such a heating rate. Most of them were based on a radially homogeneous model of Enceladus and used the Kelvin type Love numbers corresponding to this structure. Squyres et al. (1983) already remark that the rate of tidal heating (in case of homogeneous inner structure) is not sufficient to keep up the current rate of Enceladus tectonic activity. The rate of tidal heating estimated by Porco et al. (2006) is about 0.12 GW. Recently Meyer & Wisdom (2007) find that equilibrium tidal heating can not account for the observed heat released by Enceladus. Generally speaking it can be concluded that recent observations require more heat generation than current estimates based on homogeneous models provide. Data listed in Table 1 show characteristics of some moons of Saturn (together with the similar data of Io, Europa, Ganymedes, Callisto and the Earth’s moon). It is immediately evident that Enceladus has no physical properties significantly different from other moons of Saturn (except Titan), with only one exception: the observed high average density of the moon. Since its surface is formed by water ice, the high mean density proves its significant internal inhomogeneity. This circumstance results in a significantly different tidal energy distribution from that in a homogeneous model. Since homogeneous models were unable to provide sufficient heat generation by tidal friction we introduce a spherically symmetric radially heterogeneous model of Enceladus in order to see if tidal heating in such a model can explain the observed heat generation. To do so in Acceptedthe present contribution the tools ofManuscript global geodynamics are used which are shortly described in the second section of the paper. The third section describes a somewhat arbitrarily selected model of Enceladus, while the fourth describes the physics of the heating in case of the chosen model. 2. Description of elastic deformation of a spherical body In 1863 Thomson (Kelvin) (Thomson, 1863) on the basis of earlier studies of W. Hopkins derived an equation for estimation of the effective shear modulus µ of the Earth with the use of tide gauge records. On the basis of Kelvin’s work Love (1944) obtained the following expressions for the tidal deformation of a homogeneous and elastic planet 5 f 3 f 3 f g ⋅ ρ ⋅ a h = ;k = ;l = ; f = 2 f +1 2 f +1 2(2 f +1) 19μ 2 Page 2 of 11 (here h, k, l are the Love and Shida numbers g, ρ and a are the surface gravity, the mean density and the mean radius of the investigated object respectively).